Describing Translations Of Quadratic Functions A Comprehensive Guide

This article delves into the fascinating world of quadratic function transformations, providing a comprehensive guide to understanding how these transformations affect the graphs of parabolas. Specifically, we will address the question: Which phrase best describes the translation from the graph of the quadratic function $y=(x-5)^2+7$ to the graph of $y=(x+1)^2-2$? This problem involves identifying the horizontal and vertical shifts that transform one parabola into another. By understanding the vertex form of a quadratic equation and its relationship to graph translations, we can accurately determine the correct answer and gain a deeper understanding of quadratic function behavior.

Understanding Quadratic Functions and Their Graphs

To effectively tackle this problem, it's crucial to have a solid grasp of quadratic functions and their graphical representations. A quadratic function is a polynomial function of degree two, generally expressed in the form $f(x) = ax^2 + bx + c$, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards (if a > 0) or downwards (if a < 0). The most important point on a parabola is its vertex, which represents either the minimum (if the parabola opens upwards) or the maximum (if the parabola opens downwards) value of the function. Understanding how changes in the quadratic function's equation affect the parabola's position and shape is key to solving transformation problems.

The vertex form of a quadratic equation is particularly useful for understanding graph translations. The vertex form is given by $y = a(x - h)^2 + k$, where (h, k) represents the coordinates of the vertex of the parabola. The value of a determines the parabola's direction and width. A positive a means the parabola opens upwards, while a negative a means it opens downwards. The larger the absolute value of a, the narrower the parabola; the smaller the absolute value, the wider the parabola. The vertex form directly reveals the vertex of the parabola, making it easier to identify horizontal and vertical shifts.

The values of h and k in the vertex form directly correspond to horizontal and vertical translations, respectively. A change in h shifts the parabola horizontally. Specifically, replacing x with (x - h) shifts the parabola h units to the right if h is positive and h units to the left if h is negative. Similarly, a change in k shifts the parabola vertically. Adding k to the function shifts the parabola k units upwards if k is positive and k units downwards if k is negative. These transformations are fundamental to understanding how quadratic functions can be manipulated and how their graphs change as a result.

Analyzing the Given Quadratic Functions

Now, let's analyze the two quadratic functions given in the problem: $y=(x-5)^2+7$ and $y=(x+1)^2-2$. Both equations are presented in vertex form, which makes it straightforward to identify their respective vertices. For the first function, $y=(x-5)^2+7$, we can see that h = 5 and k = 7. Therefore, the vertex of this parabola is at the point (5, 7). This means that the basic parabola $y=x^2$ has been shifted 5 units to the right and 7 units upwards.

For the second function, $y=(x+1)^2-2$, we have h = -1 and k = -2. Notice that the x coordinate of the vertex is -1 because the equation includes the term (x + 1), which can be rewritten as (x - (-1)). The vertex of this parabola is at the point (-1, -2). This indicates that the basic parabola $y=x^2$ has been shifted 1 unit to the left and 2 units downwards. By identifying the vertices of both parabolas, we can determine the translation required to move from the first graph to the second graph.

The difference in the vertices will directly tell us the horizontal and vertical shifts. To move from the vertex (5, 7) to the vertex (-1, -2), we need to consider the change in both the x and y coordinates. This involves finding the difference in the x-coordinates and the difference in the y-coordinates. This process will help us understand the exact nature and magnitude of the translation, allowing us to choose the correct descriptive phrase from the given options.

Determining the Translation

To determine the translation from the graph of $y=(x-5)^2+7$ to the graph of $y=(x+1)^2-2$, we need to compare the positions of their vertices. As we established earlier, the vertex of the first parabola is (5, 7), and the vertex of the second parabola is (-1, -2). The translation involves moving from the point (5, 7) to the point (-1, -2) in the coordinate plane. To find the horizontal shift, we calculate the difference in the x-coordinates: -1 - 5 = -6. This means the graph shifts 6 units to the left. The negative sign indicates a shift in the negative x direction, which corresponds to a leftward movement.

Next, we find the vertical shift by calculating the difference in the y-coordinates: -2 - 7 = -9. This means the graph shifts 9 units down. The negative sign here indicates a shift in the negative y direction, which corresponds to a downward movement. Therefore, the translation from the graph of $y=(x-5)^2+7$ to the graph of $y=(x+1)^2-2$ involves shifting the parabola 6 units to the left and 9 units down. This comprehensive analysis of the vertex changes allows us to pinpoint the precise transformation that occurs between the two graphs.

Understanding the individual components of the translation—horizontal and vertical shifts—is essential. Each component contributes to the overall transformation of the parabola. By carefully analyzing these shifts, we can accurately describe how the graph of the quadratic function changes. This detailed approach ensures that we not only arrive at the correct answer but also develop a deeper conceptual understanding of quadratic function transformations.

Selecting the Correct Answer

Based on our analysis, the translation from the graph of $y=(x-5)^2+7$ to the graph of $y=(x+1)^2-2$ is a shift of 6 units to the left and 9 units down. Now, we can match this description with the given options:

A. 6 units left and 9 units down B. 6 units right and 9 units down C. 6 units left and 9 units up D. 6 units right and 9 units up

By comparing our findings with the options, it is clear that option A, “6 units left and 9 units down,” accurately describes the translation. The other options either incorrectly state the direction of the horizontal shift (right instead of left) or the direction of the vertical shift (up instead of down), or both. Therefore, selecting the correct answer involves a precise understanding of the shifts and their directions.

Option A correctly identifies both the direction and magnitude of the horizontal and vertical shifts. The phrase “6 units left” accurately represents the change in the x-coordinate, while “9 units down” accurately represents the change in the y-coordinate. This methodical approach of breaking down the translation into its components ensures that we arrive at the correct conclusion. By carefully analyzing each option and comparing it with our calculated shifts, we can confidently select the answer that best describes the transformation.

Conclusion

In conclusion, the phrase that best describes the translation from the graph of $y=(x-5)^2+7$ to the graph of $y=(x+1)^2-2$ is A. 6 units left and 9 units down. This problem highlights the importance of understanding the vertex form of a quadratic equation and how changes in the equation correspond to translations of the parabola's graph. By identifying the vertices of the two parabolas and calculating the horizontal and vertical shifts required to move from one vertex to the other, we can accurately describe the transformation. This approach not only helps in solving this specific problem but also builds a strong foundation for tackling more complex transformations and manipulations of quadratic functions. Understanding these concepts is crucial for success in algebra and calculus, where transformations of functions are a fundamental topic.